Recently my 2000 model Tiger developed a coolant leak from the head gasket after only 28000kms. See Truimphant or not. There are many posts on this topic some more informative then others. After some deliberation and a $1200 labour quote from my local Triumph dealer, I decided to fix it myself. It took me 15 hours work time all up which were spread over 3 weeks. This was mainly due to other commitments and some waiting for parts. The total parts cost was under $200,-
Advice No1. If you are not mechanically minded and don't want to spend the money, let it leak and keep an eye on it. The Triumph triple is a wet sleave motor which translates into: there are no oil journals in the head. It is very unlikely that water can mix with oil by virtue of a failing head gasket as there are no oil journals in the actual head to block contact surface, except the cam chain well itself (thus no preasurised oil wells). If the leak is to the outside world, chances are that's all it'll ever be.
Advice No2. If you are going to fix it: You don't have to take the motor out and my local workshop advised me not to bother supporting the engine with a jack once the head to frame bolts are removed (although the manual tells you to do just that). I didn't support it because the manual arrived after I finished the job and the engine din't fall off nor did I detect any undue stresses. Although, I have a centre stand and I kept the bike on the stand until the head frame bolts went back in. The engine is a stressed member of the frame and with it being disconnected you will notice some flex in the frame tubes (this should be ok if you don't take the bike off the stand)..
Once you have the camshafts exposed put the bike into 6th gear and rotate the engine until cylinder 1 (furthest away from the camchain) is top dead center and the cam markers are pointing toward each other, level with the block (might have to get your missus to give you a hand since you can't turn and watch easily). At this point you are ready to remove the cams and then the head. DO NOT MOVE THE MOTOR WITH THE HEAD REMOVED. The sleaves are seated into the block and sealed with silicone. If you move the pistons with the head removed you will break the seal and then you have a major major problem (that's why advice No1).
Myths: You cannot retension the head!! The tensioning procedure calls for the bolts to be undone and then tensioned in 3 stages keeping the standard tensioning pattern. Stage 1, 20Nm, stage 2, 27Nm. From here each bolt is tightened a further 90 degrees rotation. You also cannot get to the head bolts without removing the cams. If you are at that point you might as well remove the head and change the head gasket anyway.
Tips: If you park the engine as described above you need not remove the clutch cover. Simply puts the cams back in, lining up the marks and make sure all the chain slack is on the tensioner side. Your valve timing should be fine.
DON'T mix up your shims!! You will need to turn the head over to clean the mating surfaces. Cut a broom handle to the length of the cams (I actually used 3/4'' electical conduit). cover the shims with folded newspaper and clamp them in by instaling the dowels (conduit, broom handles) in place of the cams. That should allow you to turn the head upside down without loosing the shims (original tip Eurotune Queenbeyan, NSW).
Good time to check the clearences before you take the cams out and perhaps replace tight or lose shims. easy to do with the cams out. Off course recheck upon reinstallment.
Drain the cylinder block by removing that little screw in the side of the block just above the water pump. Draining at the pump doesn't drain the block.
You must clean the tops of the sleaves but be gentle. I used a corse plastic dishwashing thingo and some gumption. Came up gleaming. Don't recoment steel wool as it can scratch and leaves metal shavings. Plastic shavings will burn up if you miss some. I always clean out the cylinders with an oily rag, gently. Also stops rust if you leave the engine open for a while. Don't leave it open without covering it up with a rag or something to stop the dust falling in.
Little stumbles: The throtle cable routs on the left side, NOT with the wiring looms. It will fit either side and after 3 weeks I've forgotten.
The fuel lines are out and return. Make sure you mark them so you can put them back appropriatelly. I forgot which goes where and had to refer to some digital photos I took during disassembly. Good tip, if you have a digital camera, take lots of photos because 3 weeks is a long time to remember which plug goes where.
Good Luck